Abstract:
The effects of the A + B → C chemical reaction on miscible viscous fingering in a radial
source flow are analysed using linear stability theory and numerical simulations. This
flow and transport problem is described by a system of nonlinear partial differential
equations consisting of Darcy’s law for an incompressible fluid coupled with nonlinear
advection–diffusion–reaction equations. For an infinitely large Péclet number (Pe), the
linear stability equations are solved using spectral analysis. Further, the numerical shooting
method is used to solve the linearized equations for various values of Pe including the
limit Pe → ∞. In the linear analysis, we aim to capture various critical parameters for
the instability using the concept of asymptotic instability, i.e. in the limit τ → ∞, where
τ represents the dimensionless time. We restrict our analysis to the asymptotic limit
Da∗ (= Daτ ) → ∞ and compare the results with the non-reactive case (Da = 0) for
which Da∗ = 0, where Da is the Damköhler number. In the latter case, the dynamics
is controlled by the dimensionless parameter RPhys = −(RA − βRB). In the former case,
for a fixed value of RPhys, the dynamics is determined by the dimensionless parameter
RChem = −(RC − RB − RA). Here, β is the ratio of reactants’ initial concentration and
RA, RB and RC are the log-viscosity ratios. We perform numerical simulations of the
coupled nonlinear partial differential equations for large values of Da. The critical values
RPhys,c and RChem,c for instability decrease with Pe and they exhibit power laws in Pe.
In the asymptotic limit of infinitely large Pe they exhibit a power-law dependence on Pe
(RChem,c ∼ Pe−1/2 as Pe → ∞) in both the linear and nonlinear regimes
